Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through the center of the circle. Secant - A line, segment, or ray which intersects a circle at two points. Tangent - a line, segment, or ray which intersects a circle at one point. Point of Tangency - The point where the tangent line intersects the circle. Perpendicular to Radius Theorem - If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Converse to the perpendicular to Radius Theorem - In a plane, if a line is perpendicular to a radius of a circle at its endpoints on the circle, then the line is tangent to the circle. Tangent-Tangent Theorem - If two segments from the same exterior point are tangent to a circle, then they are congruent. 1
Developmental Proofs Given: SR is tangent to circle P ST is tangent to circle P Prove: SR ST R S P T 2
10-2 Chords and Arcs Central Angle - An angle with vertex at the center of a circle. The central angle is equal to its intercepted arc. Arc Addition Postulate - The measure of an arc formed by two adjacent arcs is the sum the measure of the two arcs. Minor Arc Congruence Theorem - In the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent. Chords Congruence Theorem - In the same circle, two chords are congruent if and only if they are equidistant from the center. Diameter Perpendicular to a Chord Theorem - If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Chord Perpendicular Bisector Theorem - If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Arc Length: The length of an arc is the length of a circle multiplied by its angular measure divided by 360. 3
Developmental Proofs 4
10-3 Inscribed Angles Inscribed angle theorem - The measure of an inscribed angle is the half the measure of its intercepted arc. Same Arc Theorem - If two inscribed angles have the same intercepted arc, then the angles congruent. Right Angle Inscribed Theorem - If an angle is inscribed in a semicircle, then it is a right angle. Quadrilateral Inscribed Theorem - The opposite angles of a quadrilateral inscribed in a circle are supplementary. Tangent Chord Theorem - The angle formed by a tangent and a chord is half the measure of the intercepted arc. 5
10-4 Circular Angle Measures Two Chords Theorem - The measure of the angle formed by two chords is half the sum of the measures of the intercepted arcs. Tangent Tangent Angle Theorem - If two lines intersect outside a circle, then the angle formed by the intersection is half the difference of the intercepted arcs. Tangent Secant Theorem - If two lines intersect outside a circle, then the angle formed by the intersection is half the difference of the intercepted arcs. Secant Secant Theorem - If two lines intersect outside a circle, then the angle formed by the intersection is half the difference of the intercepted arcs. 6
10-5 Circular Segment Measures Two Chords Intersect Theorem - If two chords intersect in a circle then the product of their segment lengths are equal. Two Secant Segment Theorem - If two secants intersect outside a circle, then the product of the secant outside the circle and the total secant length is equal to the product of the other secant outside the circle and its overall length. Tangent Secant Segment Theorem - If a secant and a tangent intersect outside of a circle, then the square of the tangent is equal to the product of the secant length outside the circle and its overall length. 7
(x - h) 2 +(y - k) 2 = r 2 10-6 Equations of Circles (h, k) is the center of the circle r is the radius of the circle 8
10-7 Areas of a Sector Area of a Sector: The area of a sector is the area of a circle multiplied by its angle measure divided by 360. 9